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In game theory, a zero-sum or constant-sum game is a game in which there is a single constant payoff that will be divided equally between the players; in order for one player to get a better result, one or more other players have to get worse results. For instance, in many games, there can be one winner and everyone else loses; there is a single "win" available. In others, you might have draws, but a draw could be considered as all of the given players splitting the win equally; for instance, in chess, a win is 1 point, a loss 0, and a draw 0.5 for each player. In some games, like poker, that constant payoff to split is 0; in order for one player to make money, another has to lose money.

A cooperative game is not be zero-sum. There are two possible outcomes; either everyone wins, or everyone loses. Thus, from the perspective of game theory, there is one outcome in which the total payoff for the players is greater than in another outcome. However, this is somewhat trivially non-zero sum; there aren't any interesting decisions that affect how much the total payoff is. Poker played at a casino, likewise, is non-zero sum as you are always losing the rake to the casino; but again, it's not in an interesting way as you don't make any interesting decisions that affect the total value of the game.

Note that all of these non-zero sum games could be made zero-sum if you consider the game itself or the house as a player; then there is one "win" split between the players and the game, or one pool of money split between the players and the casino. This is true of any non-zero-sum game; you can always posit an extra hypothetical player, or consider the interests of the house, whose gain or loss is the negative of the gain or loss of the players of the game. But it's still interesting to consider games as non-zero sum from the perspective of the players themselves.

So, are there any games which are non-zero sum in an interesting way? That is, in which players have to make interesting decisions during the game, that affect the total payoff of the game. For instance, a game in which, depending on the action of the players, the outcome could be that one player gets 10 points, or two players each get 8, or three players all get 7 points would be non-zero sum game; the total value of the game depends on decisions that the players make within the game. A game like Shadows over Camelot, which will randomly be zero-sum or non-zero sum depending on whether there's a traitor, isn't really interesting as the actions of the players don't affect what the possible payoffs are, only a random card chosen at the beginning affects this.

I am only interested in actual fun playable games, not hypothetical mathematical examples, and only interested in how games are actually played in practice, not hypothetical tournament setups that you could play; in almost any game with a variable score, you could institute a system in which your absolute score in the game was used in the tournament results, but that's not the way that most tournaments I've seen are played. I want to know about games or tournament structures that people actually play.

Note that in order for this question to even make sense, there may need to be some kind of external value associated with the game; some metagame that the game is played within, such as a tournament or ranking system, or a gambling game in which the players in an individual game may wind up with more or less money than they started with. If so, please describe how this metagame works, and interacts with the outcome of individual games.

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Republic of Rome is a potentially negative-sum game. There are multiple possible endings:

One Senator reaches 35 Influence: the player controlling that Senator wins.

One Senator is proclaimed Consul-For-Life and Rome survives to the end of the turn: The player controlling the Consul-For-Life wins.

A faction rebels and conquers Rome: That faction wins.

These are the three outright wins.

The deck runs out (effectively the time limit).

The player with the most total influence wins; this is generally regarded as a lesser win than the outright wins above.

Rome falls (four active wars, popular revolt): all players lose

This is a complete defeat, which is supposed to be regarded as worse than another player winning.

There are various tournament approaches to scoring these, but one approach is to score an outright win as +1 per other player (so +5 in a six-player game), and losing in such a game as 0, a max-influence victory as +0.5 per other player (+2.5 in a six-player) and a survival in such a game as +0.5, and all lose as -1 for each player.

I'm not sure I understood your question correctly, but my first thought is a Carcassonne tournament where the players are ranked by total score. Every game is different due to the random tile draws and the total points available in a given game varies by how people play. A tournament where multiple games are happening simultaneously leads to interesting decisions. For example, someone seeking to maximize total points might choose to make a losing move in one game if it means more points overall.

By extension, I think this includes any game with a non-deterministic amount of points (e.g., Scrabble) played more than once.

Are there actually Carcassonne or Scrabble tournaments in which your place in the tournament depends on your score within a game, and not just your score relative to the other players? If there are, that would count; you would be able to make a decision in which you lost the individual game, but wound up with a higher score and thus did better in the tournament. If so, do you have any references for such tournaments?
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Brian CampbellNov 2 '10 at 19:51

@Brian, I got the idea from a thread on BGG (boardgamegeek.com/thread/546467/…). I don't have any reference material for such a tournament, just secondhand knowledge that it exists.
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KristoNov 2 '10 at 20:19

Where in that thread does it say the tournament score is based on your score within the game, and not just your placement relative to the other players?
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Brian CampbellNov 2 '10 at 20:31

@Brian, hmm, I guess it doesn't. I must have inferred it from the emphasis the OP placed on scoring points vs. place order in each game. I can't think of any other ways to rank multiple Carcassonne games.
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KristoNov 2 '10 at 21:43

Carcassonne is non-zero-sum when multiple players share the same structure: the points are not divided between the players, yet each player receives the same amount of points. Unless off course it's not shared, but conquered.
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KonerakNov 13 '10 at 12:21

The Grognards tournaments at WBC, which are 10 classic wargames using a scoring system where you get points not just for the win (10), but you also get 2 points for each game that every opponent you beat won. This rewards players for beating good opponents, thereby keeping them from just finding new or poor players to play.

Some 3+ player games in which you can play kingmaker are implicitly non-zero-sum.

Take Risk. Let's say you make a habit of hosing anybody who dares to place in your continent, even if it's not the best move for you. In a future game, the other players will be less likely to place in your continent.

Thus, you gained an implicit advantage in future games, based on your reputation, even though you lost.

Each game's outcome thus not only yields a zero-sum account of who won and who lost, but also yields reputational factors.

There's an old game called Plank that might fit your description. When you play with three players, the player to the left of the winner receives a penalty of two points, and the third player receives a penalty of one point. When the players have played some multiple of three games, the player with the fewest points wins the match. This way, when a player has to choose between stopping one player or the other, he will always stop the player to his right.

I first ran across Plank in Sid Sackson's A Gamut of Games book. He added the rule for scoring three-player games to solve the king maker problem. Plank was described in The Young Folks' Cyclopaedia of Games and Sports in 1890 (complete rules at the link). I've since seen a commercial version called Hack Attack. I really like the game as a light filler, and I made my own set out of Lego. For me it has a similar feel to Quarto.

The way I understand your explanations I'd say many German style games should fit your description. For example in "Settlers of Catan" players gather victory points and the game ends once a player has collected a certain number of them. Since the total pool of victory points is usually far greater than the players are capable of achieving before the game ends - that should constitute a non-zero sum game. One player gaining a victory point does not significantly affect the chances of other players gaining one themself.

Similarly in "Dominion" - while play ends once the 6 points cards are depleted there are other cards that give victory points. Thus even if one player manages to grab more of the 6 point cards it is conceivable that another player wins because of additional 3 and 1 point cards.

That pattern is present in many other games, especially ones that keep all players in the game until the end (a core characteristic of German style games)

This isn't what I'm referring to. Regardless of what the total scores are, in Settlers there's a single winner, or you might look at it as 1st, 2nd, 3rd, and 4th places; either way, the outcome of the game is fixed regardless of what the actual scores are. I'm looking for a game in which the possible outcomes of the game might depend on something in-game; such as a shared win being worth two wins, instead of half a win each, or a situation in which everyone loses and actually loses points rather than getting 0. As I said, this question may only make sense in the context of a given metagame.
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Brian CampbellNov 3 '10 at 12:39

Well you could easily play a Settlers Tournament and instead of giving fixed points for 1st, 2nd etc place equate a player's tournament points to the sum of all his victory points.
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KempethNov 3 '10 at 12:53

For a few of our favourite games we maintain a high scores list. If you come a close second place then the high scores list can give you some compensation for your efforts. Also, by keeping the list of all-time low scores, we provide a bit of incentive for players who don't have a hope of winning the game.

Battle star galactica with the first expansion (Pegasus something?) should fit the bill. If you play with a Cylon leader the game has more or less three teams, humans, cylons and the leader.

The humans want to get off the planet. The cylons want to destroy them and the cylon leader draws a card to decide which side she wants to win. The tricky part is that even if she wants the humans to win she needs to hurt them a bit first to win herself. For instance destroy a certain number of fighters.
If the humans win but she hasn't destroyed enough then she still loses otherwise she wins with the humans.